Brun's constant

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Categories: Analytic number theory | Mathematical constants

In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a mathematical constant now called Brun's constant for twin primes and usually denoted by B₂ (sequence A065421 in OEIS):

Failed to parse (Missing texvc executable; please see math/README to configure.): B_2 = \left(\frac{1}{3} + \frac{1}{5}\right) + \left(\frac{1}{5} + \frac{1}{7}\right) + \left(\frac{1}{11} + \frac{1}{13}\right) + \left(\frac{1}{17} + \frac{1}{19}\right) + \left(\frac{1}{29} + \frac{1}{31}\right) + \cdots

in stark contrast to the fact that the sum of the reciprocals of all primes is divergent. Had this series diverged, we would have a proof of the twin prime conjecture. But since it converges, we do not yet know if there are infinitely many twin primes. Similarly, if it were ever to be proved that Brun's constant was irrational, the twin primes conjecture would follow immediately, whereas a proof that it is rational wouldn't decide it either way.

Brun's sieve was refined by J.B. Rosser, G. Ricci and others.

By calculating the twin primes up to 10¹⁴ (and discovering the infamous Pentium FDIV bug along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578. The best estimate to date was given by Pascal Sebah and Patrick Demichel in 2002, using all twin primes up to 10¹⁶:

B₂ ≈ 1.902160583104.

While 1.9 < B₂ is shown, no real number N is known such that B₂ < N.

There is also a Brun's constant for prime quadruplets. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by B₄, is the sum of the reciprocals of all prime quadruplets:

Failed to parse (Missing texvc executable; please see math/README to configure.): B_4 = \left(\frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13}\right) + \left(\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19}\right) + \left(\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109}\right) + \cdots

with value:

B₄ = 0.87058 83800 ± 0.00000 00005.

This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B₄. Wolf derived an estimate for the Brun-type sums B_n of 4/n. This gives the estimate for B_n of 2, about 5% higher than the true value.